As @dnidz said, you probably want to be computing the geometric mean and its CI, not the arithmetic mean and its CI, when data are sampled from a lognormal distribution. Why?

First, let's think about normal distributions. For an ideal normal population (distribution), the arithmetic mean and the median are identical. For any given sample, those two values are usually not identical. Which of the two values is likely to be closer to the true mean (and median; they are the same) of the population? The median is computed only from the ranks of the values. The arithmetic mean is computed from the actual values and takes into account an assumption about the distribution. If that assumption of sampling from a normal assumption is true, the arithmetic mean of the sample is likely to be closer to the mean and median of the distribution than is the median of the sample. That is why the arithmetic mean (the average) is so commonly used. 

Now let’s switch to the lognormal distribution. The median and geometric mean of an ideal lognormal distribution are identical, but the arithmetic mean has a larger value. How much larger depends on how asymmetrical the distribution is as expressed by the Geometric standard deviation. 
Your goal (I assume) is to estimate the median of the population (or distribution) from a sample of data. If you are sampling from a lognormal population, the best estimate is the geometric mean. That is likely to be a better estimate than the sample median and a much better estimate than the arithmetic mean. 

So it makes sense to:
 1. Transform all values to logarithms.
 2. Compute the arithmetic mean and the 95% CI of that mean from the set of logarithms.
 3. Back transform the mean and both confidence limits to the original units. You'll have calculated the geometric mean and its 95% CI.
(Biologists use the common base 10 logarithm and the 10^ back transofrm. Mathematicians and physical scientists use the natural ln logarithm and the exp() back transform. The results will be the same either way.)